MATHCOUNTS - WeeblyMATHCOUNTS Founding Sponsors: National Society of Professional Engineers,...
Transcript of MATHCOUNTS - WeeblyMATHCOUNTS Founding Sponsors: National Society of Professional Engineers,...
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2013State Competition
Sprint RoundProblems 1–30
Total Correct Scorer’s Initials
Copyright MATHCOUNTS, Inc. 2013. All rights reserved.
MATHCOUNTS ®
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.
This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. If you are wearing a calculator wrist watch, please give it to your proctor now. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.
In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.
03-S13SPR
HONOR PLEDGE I pledge to uphold the highest principles of honesty and integrity as a Mathlete®. I will neither give nor accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand that any competitor found to be in violation of this honor pledge is subject to disqualification.
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National Sponsors Raytheon Company * Northrop Grumman Foundation * U.S. Department of Defense * National Society of Professional Engineers * CNA Foundation * ConocoPhillips * Texas Instruments Incorporated * 3M Foundation * Art of Problem Solving * NextThought
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round
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Ifthesumoftwonumbersis4andtheirdifferenceis2,whatistheirproduct?
MaryandAnnliveatoppositeendsofthesameroad.Theyplantoleavehomeatthesametimeandridetheirbikestomeetsomewherebetweenthetwohouses.At11:00a.m.Maryhastraveledhalfofthedistancebetweentheirhouses.Annisridingmoreslowlyandhascoveredonly
38ofthedistance
betweenthehouses.Theyarestillonemileapart.Howmanymilesapartaretheirhouses?
Abagcontains8bluemarbles,4redmarblesand3greenmarbles.Inasingledraw,whatistheprobabilityofnotdrawingagreenmarble?Expressyouranswerasacommonfraction.
Thearithmeticmeanof11numbersis78.If1issubtractedfromthefirst,2issubtractedfromthesecond,3issubtractedfromthethird,andsoforth,until11issubtractedfromtheeleventh,whatisthearithmeticmeanofthe11resultingnumbers?
Anoptometristhasthislogoonhisstorefront.Thecentercirclehasarea36πin2,anditistangenttoeachcrescentatitswidestpoint(AandB).TheshortestdistancefromAtotheoutercircleis
13 thediameterofthesmaller
circle.Whatistheareaofthelargercircle?Expressyouranswerintermsofπ.
in2
miles
A
B
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round
Excludingsalestax,howmuchwillDorissavewhenshebuysaDVDoriginallypricedat$12.00andnowonsalefor20%off ?
Whatisthevalueof2 2444 111
444 111−− ?
Theproductofthedigitsofpositiveintegern is20,andthesumofthedigitsis13.Whatisthesmallestpossiblevalueofn?
QuadrilateralABCDisasquarewithBC=12cm.BOC andDOC aresemicircles.Intermsofπ,whatistheareaoftheshadedregion?
Realnumbersaandbsatisfytheequation 2 4 3 15 5− +
+ =a a b .Whatisthe
valueofa−b?Expressyouranswerasacommonfraction.
Ifthepoint(x,x)isequidistantfrom(−2,5)and(3,−2),whatisthevalueofx?
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cm2
O
C
B
D
A
12
$
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round
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Inabagofmarbles,25 ofthemarblesarered,
310 ofthemarblesare
whiteand110 ofthemarblesareblue.Iftheremaining10marbles
aregreen,howmanymarblesareinthebag?
Iftis40%greaterthanp,andpis40%lessthan600,whatisthevalueoft–p?
Howmanywayscanallsixnumbersintheset{4,3,2,12,1,6}beorderedsothatacomesbeforebwheneveraisadivisorofb?
Whatistheunitsdigitoftheproduct723×8105×318?
If4(a–3)–2(b+5)=14and5b–a=0,whatisthevalueofa+b?
ThetwoconesshownhaveparallelbasesandcommonapexT.TW=32m,WV=8mandZY=5m.WhatisthevolumeofthefrustumwithcircleWandcircleZasitsbases?Expressyouranswerintermsofπ.
Acoinisflippeduntilithaseitherlandedheadstwotimesortailstwotimes,notnecessarilyinarow.Ifthefirstfliplandsheads,whatistheprobabilitythatasecondheadoccursbeforetwotails?Expressyouranswerasacommonfraction.
marbles
m3
ways
T
YZX
VWU
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round
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Theproductoftwoconsecutiveintegersisfivemorethantheirsum.Whatisthesmallestpossiblesumoftwosuchconsecutiveintegers?
Fournickels,onepennyandonedimeweredividedamongthreepiggybankssothateachbankreceivedtwocoins.Labelsindicatingtheamountineachbankweremade(6cents,10centsand15cents),butwhenthelabelswereputonthebanks,nobankhadthecorrectlabelattached.Sorayashookthepiggybanklabeledas15cents,andoutfellapenny.Whatwastheactualcombinedvalueofthetwocoinscontainedinthepiggybankthatwaslabeled6cents?
Supposethe9×9multiplicationgrid,shownhere,werefilledincompletely.Whatwouldbethesumofthe81products?
Insomelanguages,everyconsonantmustbefollowedbyavowel.Howmanyseven-letter‟words”canbemadefromtheHawaiianwordMAKAALAifeachconsonantmustbefollowedbyavowel?
Iff(x)=3x2,whatisthex-coordinateofthepointofintersectionofthegraphsofy=f(x)andy=f(x−4)?
InisoscelestrapezoidABCD,shownhere,AB=4unitsandCD=10units.PointsEandFareonCD with BE paralleltoAD andAF paralleltoBC .AF andBE intersectatpointG.WhatistheratiooftheareaoftriangleEFGtotheareaoftrapezoidABCD?Expressyouranswerasacommonfraction.
‟words”
A B
CDE F
G
cents
× 1 2 3 4 5 6 7 8 9123456789
14
12
40
6¢ 10¢
15¢
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Copyright MATHCOUNTS, Inc. 2012. All rights reserved. 2013 State Sprint Round
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Thesumoffiveconsecutive,positiveevenintegersisaperfectsquare.Whatisthesmallestpossibleintegerthatcouldbetheleastofthesefiveintegers?
If123+125+127+129+12x=1011102,whatisthevalueofx,thebaseofthefifthterm?
Aboxcontains r redballsandggreenballs.Whenrmoreredballsareaddedtothebox,theprobabilityofdrawingaredballatrandomfromtheboxincreasesby25%.Whatwastheprobabilityofrandomlydrawingaredballfromtheboxoriginally?Expressyouranswerasacommonfraction.
ThegameofConnexcontainsone4-unitpiece,twoidentical3-unitpieces,threeidentical2-unitpiecesandfouridentical1-unitpieces.Howmanydifferentarrangementsofpieceswillmakea10-unitsegment?The10-unitsegmentsconsistingofthepieces4-3-2-1and1-2-3-4aretwosucharrangementstoinclude.
Insquareunits,whatisthelargestpossibleareaarectangleinscribedinthetriangleshownherecanhave?
AlinesegmentwithendpointsA(3,1)andB(2,4)isrotatedaboutapointintheplanesothatitsendpointsaremovedtoA' (4,2)andB' (7,3),respectively.Whatarethecoordinatesofthecenterofrotation?Expressyouranswerasanorderedpair.
10 17
21
units2
(,)
arrange-ments
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